Book 11
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If two parallel planes be cut by any plane, their common sections are parallel.
16
For let the two parallel planes AB, CD be cut by the plane EFGH, and let EF, GH be their common sections; I say that EF is parallel to GH.
16
For, if not, EF, GH will, when produced, meet either in the direction of F, H or of E, G.
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Let them be produced, as in the direction of F, H, and let them, first, meet at K.
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Now, since EFK is in the plane AB, therefore all the points on EFK are also in the plane AB. [XI. 1]
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But K is one of the points on the straight line EFK; therefore K is in the plane AB.
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For the same reason K is also in the plane CD; therefore the planes AB, CD will meet when produced.
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But they do not meet, because they are, by hypothesis, parallel; therefore the straight lines EF, GH will not meet when produced in the direction of F, H.
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Similarly we can prove that neither will the straight lines EF, GH meet when produced in the direction of E, G.
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But straight lines which do not meet in either direction are parallel. [I. Def. 23]
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Therefore EF is parallel to GH.
16
Therefore etc. Q. E. D.
PROPOSITION 17.
17
If two straight lines be cut by parallel planes, they will be cut in the same ratios.
17
For let the two straight lines AB, CD be cut by the parallel planes GH, KL, MN at the points A, E, B and C, F, D; I say that, as the straight line AE is to EB, so is CF to FD.
17
For let AC, BD, AD be joined, let AD meet the plane KL at the point O, and let EO, OF be joined.
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Now, since the two parallel planes KL, MN are cut by the plane EBDO, their common sections EO, BD are parallel. [XI. 16]
17
For the same reason, since the two parallel planes GH, KL are cut by the plane AOFC, their common sections AC, OF are parallel. [id.]
17
And, since the straight line EO has been drawn parallel to BD, one of the sides of the triangle ABD, therefore, proportionally, as AE is to EB, so is AO to OD. [VI. 2]
